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# What is the tens digit of 7^1415?

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Math Expert
Joined: 02 Sep 2009
Posts: 56306
What is the tens digit of 7^1415?  [#permalink]

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25 Apr 2016, 04:08
1
11
00:00

Difficulty:

65% (hard)

Question Stats:

55% (01:52) correct 45% (01:51) wrong based on 248 sessions

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What is the tens digit of 7^1415?

A. 0
B. 1
C. 2
D. 3
E. 4

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Math Expert
Joined: 02 Aug 2009
Posts: 7763
What is the tens digit of 7^1415?  [#permalink]

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25 Apr 2016, 06:18
2
2
Bunuel wrote:
What is the tens digit of 7^1415?

A. 0
B. 1
C. 2
D. 3
E. 4

Hi,
Another approach to this Q, which would be correct for all numbers..

the Q can be answered if we can find the remainder when $$7^{1415}$$ is divided by 100..
now 7^4=2401..

$$(7^4)^{354} * 7^3 = (2401)^{354} *7^3$$..
2401will always leave a remainder of 1
so finally remainder = $$1* 7^3 = 1* 343$$..
or remainder = 43..
tens digit =4
E
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Re: What is the tens digit of 7^1415?  [#permalink]

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25 Apr 2016, 06:05
5
3
7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2401
7^5 = 16807
7^6 = 117649
We should see this as pattern recognition . We have a cycle of 4 . (We can multiply the last 2 digits only as we care about ten's digit )
0 , 4 , 4 , 0 .
1415= 4*353 + 3
The ten's digit will be 4 .
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##### General Discussion
Manager
Joined: 11 Jul 2016
Posts: 78
Re: What is the tens digit of 7^1415?  [#permalink]

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09 Aug 2016, 10:54
Skywalker18 wrote:
7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2401
7^5 = 16807
7^6 = 117649
We should see this as pattern recognition . We have a cycle of 4 . (We can multiply the last 2 digits only as we care about ten's digit )
0 , 4 , 4 , 0 .
1415= 4*353 + 3
The ten's digit will be 4 .

I am bit confused here

Clearly the cyclicity is 4 here.

7^1415

= 7 ^[4.353 + 3]
= (7 ^4.353) . (7 ^3)
= (7^4.(353)) . (7^3)
= (2401^353) . ( 7^3 )
= (Tens digit of a number that ends in 1 is equal to the last digit of power multiplied by the tens digit of the number) .(7^3)
= (0) . (343)
= 0
= The tens digit is 0.
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Re: What is the tens digit of 7^1415?  [#permalink]

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10 Aug 2016, 04:09
Manonamission wrote:
Skywalker18 wrote:
7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2401
7^5 = 16807
7^6 = 117649
We should see this as pattern recognition . We have a cycle of 4 . (We can multiply the last 2 digits only as we care about ten's digit )
0 , 4 , 4 , 0 .
1415= 4*353 + 3
The ten's digit will be 4 .

I am bit confused here

Clearly the cyclicity is 4 here.

7^1415

= 7 ^[4.353 + 3]
= (7 ^4.353) . (7 ^3)
= (7^4.(353)) . (7^3)
= (2401^353) . ( 7^3 )
= (Tens digit of a number that ends in 1 is equal to the last digit of power multiplied by the tens digit of the number) .(7^3)
= (0) . (343)
= 0
= The tens digit is 0.

Yes, the cyclicity is 4 but considering that we start from the exponent of 1, the cyclicity is
0, 4, 4, 0
0, 4, 4, 0
...
etc

So 1415 = 4*353 + 3 means 353 full cycles and 3 more so you go to 0, 4, 4
The digit will be 4.
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What is the tens digit of 7^1415?  [#permalink]

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01 Nov 2016, 02:46
1
$$7^{1415}$$ = $$7^{1410}*7$$ = $$49^{705}*7$$ = $$49^{704}*49$$ = $$01^{352}*49$$ = $$1*49=49$$
Tens digit is 4
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Joined: 17 Aug 2016
Posts: 48
Re: What is the tens digit of 7^1415?  [#permalink]

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01 Nov 2016, 15:29
chetan2u wrote:
Bunuel wrote:
What is the tens digit of 7^1415?

A. 0
B. 1
C. 2
D. 3
E. 4

Hi,
Another approach to this Q, which would be correct for all numbers..

the Q can be answered if we can find the remainder when $$7^{1415}$$ is divided by 100..
now 7^4=2401..

$$(7^4)^{354} * 7^3 = (2401)^{354} *7^3$$..
2401will always leave a remainder of 1
so finally remainder = $$1* 7^3 = 1* 343$$..
or remainder = 43..
tens digit =4
E

Hi chetan2u, sorry, but why you are calculating (7^4)^{354} * 7^3? Shouldn't it be (7^4)^{353} * 7^3? Or am I missing something??
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Posts: 1363
Location: India
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What is the tens digit of 7^1415?  [#permalink]

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25 Dec 2018, 02:11
1

I adore chetan2u 's approach above. Is there a way to skip two digit multiplications since
they are too prone to error?
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What is the tens digit of 7^1415?   [#permalink] 25 Dec 2018, 02:11
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